# Extended-MLE computation for sparse ZIP seasonal effects

During the first revision run, all general-outcome models and the core-DV nonseasonal models converged, but core-DV R3 developed a singular information matrix as ZIP seasonal coefficients diverged. This failure was retained and investigated; it was not resolved by relaxing tolerances, penalizing coefficients or selecting a result.

Independent review identified two ZIPs, 92011 and 92110, each with one positive core-DV record group. Each positive outcome lies at a single annual phase. For those ZIPs, the intercept/sine/cosine direction cos(phase − positive_phase) − 1 is zero wherever the outcome is positive and nonpositive at every other row. It is strictly negative on 2,918 zero-count rows in the nonstation sample. Thus these zeros have fitted mean zero at the extended maximum-likelihood boundary; a finite unrestricted seasonal nuisance estimate does not exist.

The runner now certifies this direction for every outcome and annual specification. It removes only certified zero-count rows, retains all positive outcomes and every matching-phase zero, and removes the two seasonal columns now redundant with the ZIP intercept. This computes the identifiable interior coefficients of the same planned Poisson mean model at its extended-MLE boundary. It is not a minimum-event cutoff, a new temperature hypothesis or a ridge prior. The retained sample, removed zero rows, redundant columns and certificates are exported for each fit. Full descriptive denominators remain unchanged. Station-paired models use the same certification on their common sample. Calendar score gaps remain explicit.

The implementation still requires a converged finite score solution after this reduction; additional unhandled separation must fail visibly. The existing 24-test family remains unchanged. This numerical amendment followed a diagnosed failure and is not presented as a choice made before the first revision results were known. Independent validation must check every certificate and the equivalence of the reduced spline basis and covariance.